Transcript Linear Systems

Linear Systems
What is the
Matrix?
Outline
• Announcements:
– Homework III: due Wed. by 5, by e-mail
• Office Hours: Today & tomorrow, 11-1
– Ideas for Friday?
• Linear Systems Basics
• Matlab and Linear Algebra
Ecology of Linear Systems
• Linear Systems are found in every
habitat:
– simple algebra
– solutions to ODEs & PDEs
– statistics (especially, least squares)
• If you can formulate your problem as
linear system, it will be easy to solve on
a computer
“Standard Linear System”
• Simplest linear system: finding the equation of
a line:
– y=m*x + b
• The goal is to find m and b from observations
of (x,y) pairs
• 2 points form a line, so we need two
observations (x1, y1) & (x2,y2)
The 8th Grade Solution
• Solve for b in the first equation:
• Substitute this for b in 2nd equation &
solve for m:
• Put this into the equation for b above
The Sophomoric Solution
• Write the equations as a matrix problem
• Perform Gaussian Elimination on Matrix:
Comparing methods
• Gaussian Elimination is a simpler
algorithm
– Easily generalizes to systems with more
unknowns
• Gaussian Elimination is the starting
point of much of numerical linear
algebra
A Closer Look at GE
(optional)
• For Am=y
– GE reduces A to an upper triangular matrix
– Perform “back substitution” using modified
y
– Modified y is equivalent to Ly
• L is a lower triangular matrix
• A=L*U
Other Algorithms for Solving
Linear Systems
• GE aka LU decomposition -- any A
• Cholesky Factorization -- symmetric,
positive definite A
• Iterative solvers (conjugate gradients,
GMRES)
Linear Systems in Matlab
• Linear systems are easy in Matlab
– To solve Ax=b, type x=A\b
– To solve x’A’=b’, type x’=b’/A’ (transposed)
More About \
• Matrix multiplication (*) is easy, fast
• Matrix “division” (\) is hard and
computationally intensive
– In general, performs GE with partial
pivoting
– But, \ is smart & looks closely at A for
opportunities to speed up
• If A is LT, just does back substitution
• If A is over-determined, A\b is the
least-squares solution
Factorization
• Can explicitly factor A using LU:
– [L,U]=lu(A)
– useful if you have to solve A\bmany times
(different b each time)
• To solve LUx=b: first solve Ly=b, then solve Ux=y
• In Matlab: y=L\b; x=U\y;
• Other factorizations: chol, svd
What about A-1?
• Matlab can compute A-1 using inv(A), but …
– inv(A) is slower than lu(A)
– There are numerical problems with inv(A)
• Rarely needed, use lu(A) or another
factorization